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STATISTIKA INDUSTRI 2

TIN 4004

Pertemuan 9

• Outline: – Multiple Linear Regression and Correlation – Non Linear Regression

• Referensi:

– Montgomery, D.C., Runger, G.C., Applied Statistic and Probability for Engineers, 5th Ed. John Wiley & Sons, Inc., 2011.

– Walpole, R.E., Myers, R.H., Myers, S.L., Ye, K., Probability & Statistics for Engineers & Scientists , 9th Ed. Prentice Hall, 2012.

Multiple Linear Regression

• Terdiri atas lebih dari satu independent variable

• Metode yang digunakan untuk estimasi koefisien: – Least square estimation (metode kuadarat

terkecil)

– Normal equation (Persamaan Normal)

– Matrix approach (Sistem Matriks)

Multiple Linear Regression

• Terdiri atas lebih dari satu independent variable

• Metode yang digunakan untuk estimasi koefisien: – Least square estimation (metode kuadarat

terkecil)

– Normal equation (Persamaan Normal)

– Matrix approach (Sistem Matriks)

Multiple Linear Regression Penentuan Koefisien

• Least square estimation (metode kuadarat terkecil)

Multiple Linear Regression Least square estimation

• Persamaan least square:

• Least square normal equations:

LEAST SQUARE ESTIMATOR

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Multiple Linear Regression Least square estimation

• Contoh soal:

Multiple Linear Regression Least square estimation

Multiple Linear Regression Least square estimation

• Contoh soal:

Multiple Linear Regression Penentuan Koefisien

• Matrix approach (Sistem Matriks)

Model umum:

Normal Equations :

Least square Estimate of β :

Multiple Linear Regression Matrix approach

p = k + 1 p x p p x 1 p x 1

• Contoh soal

Multiple Linear Regression Matrix approach

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• Contoh soal

Multiple Linear Regression Matrix approach

Multiple Linear Regression Estimator of Variance

• Residual:

– the difference between the observation 𝑦𝑖 dengan nilai 𝑦 𝑖

–

Multiple Linear Regression Estimator of Variance

• Residual:

– Contoh soal:

Multiple Linear Regression Estimator of Variance

• Variance Estimator

Error atau Residual Sum of Squares

Multiple Linear Regression Estimator of Variance

• Variance Estimator

Contoh soal:

𝜎 2 = 𝑠2 =? ? ? ?

Multiple Linear Regression Uji Hipotesa

• Uji Nilai Individu Koefisien Regresi

– Area Penolakan:

Partial / Marginal Test

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Multiple Linear Regression Uji Hipotesa

– Contoh Soal:

– Kesimpulan: ???

Multiple Linear Regression Uji Hipotesa

• Uji Nilai Individu Koefisien Regresi

– Area Penolakan:

Partial / Marginal Test

Multiple Linear Regression Uji Hipotesa

– Contoh Soal:

– Kesimpulan: ???

Multiple Linear Regression Uji Hipotesa

• Uji Kesesuaian Model (Fitted Model Hypothesis Testing) – The ability of the entire function to predict the

true response in the range of the variables considered

– Reject 𝐻0interpret at least one regressor variable contributes significantly to the model

Multiple Linear Regression Uji Hipotesa

• Uji Kesesuaian Model (Fitted Model Hypothesis Testing)

– Menggunakan uji F

– Area Penolakan: 𝒇 > 𝒇𝜶(𝒗𝟏=𝒌,𝒗𝟐=𝒏−𝒑)

𝒇 =𝑺𝑺𝑹/𝒌

𝑺𝑺𝑬/(𝒏 − 𝒑)=

𝑺𝑺𝑹/𝒌

𝒔𝟐

Multiple Linear Regression Uji Hipotesa

• Uji Kesesuaian Model (Fitted Model Hypothesis Testing)

– Format ANOVA

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Multiple Linear Regression Confident Interval

• CI on Individual Regression Coefficients

– Contoh:

Multiple Linear Regression Confident Interval

• CI on Mean Response

– Contoh:

Multiple Linear Regression Correlation

• Coefficient of multiple determination 𝑅2

• Adjoint 𝑅2

Multiple Linear Regression Multicollinearity

• strong dependencies among regressor variables 𝑥𝑗 – The estimates of the regression coefficients are

very imprecise and affects the stability of the regression coefficients.

– To detect: • Variance inflation factors > 1

• Significant F-test of significance of regression, but tests on the individual regression coefficients are not significant

Multiple Linear Regression Uji Hipotesa

• Uji Koefisien Subset

– Test the siginificance of a set of variables. Test contribution of new variables.

– Menggunakan uji F

Partial F-test

– Area Penolakan: 𝒇 > 𝒇𝜶(𝒗𝟏=𝒓,𝒗𝟐=𝒏−𝒑)

𝒇 =𝑺𝑺𝑹(𝜷𝒋|𝜷𝟎, 𝜷𝟏, … , 𝜷𝒋−𝟏, 𝜷𝒋+𝟏, … , 𝜷𝒌)/𝒓

𝑺𝑺𝑬/(𝒏 − 𝒑)

=(𝑺𝑺𝑹 𝜷𝟏,𝜷𝟐,…,𝜷𝒌 𝜷𝟎 −𝑺𝑺𝑹 𝜷𝒋 𝜷𝟎 )/𝒓

𝒔𝟐

Multiple Linear Regression Uji Hipotesa

• Uji Koefisien Subset

– Contoh: Kasus Wire Bond Strength

𝒇 =𝟑𝟑. 𝟐/𝟐

𝟒. 𝟏= 𝟒. 𝟎𝟓

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NOTE: partial F-test to a single variable = t-test

General Linear Model (GLM)

• GLM is the mathematical framework used in many common statistical analysis, including multiple regression and ANOVA

– ANOVA is typically prsented as distinct from multiple regression but it IS a multiple regression

Characteristics of GLM

• Linear, pairs of variables are assumed to have linear relations

• Additive, if one set of variables predict another variable, the effect are thought to be additive

• BUT! This does not preclude testing non-linear or non additive effects (by doing some transformations)

Analysis of Variance (ANOVA)

• Appropriate when the predictors (independent variables) are all categorical and the outcome (dependent variable) is continous – Most common application is to analyze data from

randomized experiments

• More specifically, randomized experiments that generate more than 2 means – If only 2 means thes use:

• Independent t-test

• Dependent (paired) t-test

NONLINEAR REGRESSION

Nonlinear Regression

Beberapa Jenis Nonlinear Regression:

• Polynomial Regression Models

– Bersifat curvilinear

• Logistic Regression

– For non normal distribution data, binary responses

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TUGAS KELOMPOK

• Cari kasus permasalahan yang diselesaikan dengan: – One way ANOVA

– Factorial ANOVA

– Simple Linear Regression

– Multiple Linear Regression

• Selesaikan dengan menggunakan software statistik

• Interpretasikan hasil output software tersebut

• Catatan: – Kasus yang digunakan tidak boleh sama antar kelompok

– Tugas dipresentasikan pada pertemuan selanjutnya

Pertemuan 10 - Persiapan

• Materi Presentasi Tugas – ANOVA dan Regresi Linier: Software dan aplikasi